Enhancing the extended hensel construction by using Gröbner basis

Contrary to that the general Hensel construction (GHC: [3]) uses univariate initial Hensel factors, the extended Hensel construction (EHC: [8]) uses multivariate initial Hensel factors determined by the Newton polygon (see below) of the given multivariate polynomial F (x, u) ∈ K[x, u], where (u) = (u₁,...,u_ℓ), with ℓ ≥ 2, and K is a number field. The F(x, u) may be such that its leading coefficient may vanish at (u) = (0) = (0,...,0), and even may be F(x, 0) = 0. The EHC was used so far for computing series expansion of multivariate algebraic function determined by F(x, u) = 0, at critical points [8, 5] and for factorization [4, 1] and GCD computation [7] of F(x, u), without shifting the origin of u. It allows us to construct efficient algorithms for sparse multivariate polynomials [1, 7]. The EHC is another and promising approach than Zippel's sparse Hensel lifting [9, 10].